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This paper aims to investigate numerical approximation of a general secondorder non-autonomous semilinear parabolic stochastic partial differentialequation (SPDE) driven by multiplicative noise. Numerical approximations ofautonomous SPDEs are thoroughly investigated in the literature, while thenon-autonomous case is not yet understood. We discretize the non-autonomousSPDE driven by multiplicative noise by the finite element method in space andthe Magnus-type integrator in time. We provide a strong convergence proof ofthe fully discrete scheme toward the mild solution in the root-mean-square$L^2$ norm. The result reveals how the convergence orders in both space andtime depend on the regularity of the noise and the initial data. In particular,for multiplicative trace class noise we achieve convergence order$\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+\Deltat^{1/2}\right)$.  Numerical simulations to illustrate our theoretical finding are provided.

Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise